Real-rooted atlas

A145324 → basis A013611 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(6 + 4\,x + 5\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
111819
217626096532
3369682766176153619684
49609628909632609616345630720905464
52978976112682801704728012893440487526473728049800520
6107243136489069408929213920941487680536525824163049472206438403187233280
7439696857623483626080537479917766833584032052125164544238538547206063980544660602880232668029440
82022605544961238537668416331781816057650783763586564857847820544297378324070411376818749442486910320642378170368019078778414080
9103152882792967125584326905621875023292601639170992071449645088529319398434597685809766417697310525030458190517633024111605081702409512681472001736168835681280
105776561436405764444199907356160153852701476953603156076580875648042484812930300928392136569540044802513347565031424011045485618462720318537123338649654433337573376041855798476800173616883568128000
113523702476207513629882368924347340811518230745625126402663699681423522560410649034756866764844313040948487178243415397548461383680188018145394130944072449095492278681618610215512231116828681205742305280200907832688640018924240308925952000
1223256436342969589762155468877769715200091559154761786451456235698978851203146240409540746373555237888506003568333586094080455844051130584649728301692648480118374400145585838629852020736499562718940574515201157027164535232921616240218681743769601044720729980928002233060356453262336000
1316512069803508408729616606189467371274946567707762559637878565376218619401650954646645764227653580818979807948858860535148603706001408607011274569523313991684694744490540114496716827231382857606122438656116997022677497922191363619040512887232659456763240764780111200256983627418006296985605850436087893196800283598665269564316672000
1412549173050666390634496136114281834126734831616685427091336706420647936212397320612552802843033645247331313281525339156487009992819785269540876288814491779564459954642124872100734602276269014548484886431791502134003367936252306034380535155497369697702921504441721526681627514872889652241078681653270014263654529105920634639765071766487040035102616527359180800038569418476660747067392000
Technical details
Node id
n-4c1ffa8a6e81d9b0 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-1a6646c65534e500

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:              1;
  1:             11,              8;
  2:            176,            260,              96;
  3:           3696,           8276,            6176,            1536;
  4:          96096,         289096,          326096,          163456,           30720;
  5:        2978976,       11268280,        17047280,        12893440,         4875264,          737280;
  6:      107243136,      489069408,       929213920,       941487680,       536525824,       163049472,        20643840;
  7:     4396968576,    23483626080,     53747991776,     68335840320,     52125164544,     23853854720,      6063980544,      660602880;
  8:   202260554496,  1238537668416,   3317818160576,   5078376358656,   4857847820544,   2973783240704,   1137681874944,   248691032064,    23781703680;
  9: 10315288279296, 71255843269056, 218750232926016, 391709920714496, 450885293193984, 345976858097664, 176973105250304, 58190517633024, 11160508170240, 951268147200;
%F Formula
The row polynomials satisfy P(n,x) = (5*n+6+(4*n+4)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(5*n + 6 + (4*n + 4)*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (5*n + 6 + (4*n + 4)*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.